For each \(k\ge 1\) , consider a distribution \(\bar{\nu }_k\)  on \(\Delta _k\)  and corresponding random probability
vectors \((\varepsilon _1^{(k)}, \dots , \varepsilon _k^{(k)})\)  driving the coagulation dynamics when the population
size drops down to \(k\) . Also let \(\bar{c}_k=\mathbb {E}\left[\sum _{i=1}^k \left(\varepsilon _i^{(k)}\right)^2\right]\)  and consider
a measure \(F\)  on \(\mathbb {N}\)  satisfying
\(0<\sum F(k)\bar{c}_k<\infty \)  (a necessary condition for (REF ) below to indeed be a coagulation measure, see
section 4.2.4 in [1]}).
